Computing apparatus



157 R. E. SPENCER Ei-AL 2,781,9fi?

COMPUTING APPARATUS Filed Jan. 25, 19511 2 Sheets-Sheet 1 1 00 FIG. 1

f/ Mwnfrs:

ROLF EDMUND SPENCER RICHARD HERBERT BOOTH Feb. 19, 1957 R. E. SPENCER ETAL 2,781,967

' COMPUTING APPARATUS Filed Jan. 25, 1951 2 Sheets-Sheet 2 /nven72 rs: ROLF EDMUND SPENCER RICHARD HERBERT BOOTH COMPUTING APPARATUS Rolf Edmund Spencer, Ealing, London, and Richard Herbert Booth, Beaconsfield, England, assignors to Electric I & Musical Industries Limited, Hayes, England, a company of Great Britain Application January 25, 1951, Serial No. 207,728

Claims priority, application Great Britain January 28, 1950 11 Claims. (Cl. 235-61) This invention relates to computing apparatus. In the United States Patent application Serial No.

110,404, filed August 15, 1949, by R. E. Spencer and G. H; Stephenson, now forfeited, there is described computing apparatus adapted to evaluate an arbitrary function by utilising Taylors expansion for the function. The 0 apparatus involves storing selected particular values of the function and one or more of its derivatives, by means of digital methods.

The object of the present invention is to provide im- 5 proved computing apparatus with a view to simplifying the construction thereof.

Y According to the present invention there is provided computing apparatus adapted to evaluate a function by utilising Taylors expansion for the function and wherein derivatives of the function employed in the expansion are automatically evaluated by the apparatus from values .of the function, which may be either stored in the apparatus or evaluated thereby.

In one form of the present invention the derivatives may be automatically evaluated from the stored selected values of the function, and this form of the invention is applicable to evaluating functions such as exponential functions and transcendental functions where the derivatives are simply related to the function itself, being for example equal to the function multiplied by a numerical constant. p

In another and preferred form of the present invention derivatives employed in the expansion are derived by feedback from the output of the computing apparatus. This form of the invention makes use of the fact that, as will hereinafter appear, it is possible to modify Taylors ex: Pension and express it in the form f(x) =f(x--6x) It will be observed that in this form of Taylors expansion the terms which involve derivatives are expressed in terms of the value of f(x) evaluated by the apparatus so that it is possible to use feedback from the output thereof.

According to one application of this latter form of the present invention there is provided apparatus for evaluating the instantaneous value of cos x and sin x.

According to another application of this latter form of the present invention there is provided apparatus for converting Cartesian co-ordinates x and y into polar coordinates -R and S, comprising means for settingup a quantity it cos 6+y sin 0 and a quantity x sin 0-y cos 0, and means for varying 0 to make the second-mentioned 6 quantity substantially zero thereby rendering 9 equal .to S and the first-mentioned quantity equal to R.

.111 order that the present invention may be clearly understood and readily carried into effect the same will now be more fully described with reference to the accompanying drawings in which:

Figure 1 illustrates diagrammatically one application i a. Patented Feb. 19,

of the present invention to the evaluation of the exponential function,

Figure 2 illustrates diagrammatically an application of the present invention to the evaluation of transcendental functions, 1

Figure 3 illustrates diagrammatically an application of the present invention to the de-resolution of Cartesian co-ordinates into polar co-ordinates, and a Figure 4 illustrates a. modification of the apparatus illustrated in Figure 3.

Referring to the drawings, the simple apparatus illustrated in Figure 1 illustrates in principle the application of the present invention in the first-mentioned form to the synthesis of exponential and trigonometrical functions. In the case of the exponential function In Figure l voltages representative of successive particular values of f(x) are set up at the studs bob]. 171' of a low speed switch. The studs are connected to taps on a potentiometer 1 fed from a reference voltage source 2. The tapped potentiometer 1 can thus be regarded as means for'storing values of the function for successive particular values of the variable. The selector member 3 of the low speed switch has two contact brushes 4 and 5 and is suitably controlled so that its displacement represents the instantaneous value of x. An interpolating potentiometer 6' is fed with a voltage equal to the volt. age set up at that one of the studs b0 hr with which the member 3 is at the time in engagement, multiplied by the constant a. A suitable high speed switch not shown is provided to connect the potentiometer 6 alternately to the brushes 4 and 5 in dependence upon the order in which the brushes engage the contacts b0 hr. The potentiometer 6 has two high speed brushes 7 and 8 connected respectively to the brushes 4 and 5 and driver.

7 synchronously therewith so that each scans the potentiometer 6 While the corresponding brush 4 or -5 moves through the range of positions in which it remains in engagement with the respective stud of the series he br- The.displacement of the brush 7 or the brush 8 represents the instantaneous value of 6x. The output is obtained at 9and represents (x )+af(x )8x. This is approximately equal to ';f(x,-,+6x)' so that the apparatus illustrated can be applied to evaluate approximately f(x) for any value of x between x and x,. The principle illustrated. could of course be applied also to the interpolation of the quadratic and higher order terms in Taylors expansion for f(x) wheremore accurate evaluation of f(x) is required or where larger intervals between the studs bobl b1 are used.-

In the case of sine and cosine functions,

a sin each function (apart from constant terms) being 'the positive of alternat'e derivatives of the other function.

The derivatives can be' obtained over a range of by providing an'extension of the studs bobr [71' to cover a further right angle, the studs being interconnectedlso' 5 that'a s'tud'in position will be joined to thatin position" If the first set of studs is then connected to voltages representing sin cp a pair of brushes 90 apart will pickup voltages corresponding to sin 5 and cos 1p over the interval of 0-90, which are the alternate interpolation coefficients required for utilising Taylor-s expansion-to evaluate f(x). Since in many cases both sine and cosine functions will be required the second brush would be already provided apart from the need for both functions for interpolation purposes. In the following description no further reference is made to a constant term such as a since it will be appreciated that provision can readily be made for it, as a constant of proportionality, in the initial construction of the machine. It is a disadvantage of the method above that high speed switching is required to change between two slow speed brushes (e. g. 4 and for feeding interpolation otentiometers (e. g. 6). This can be overcome by using the final output from the unit, f(x), instead of the stored particular values of f(x), say f(x as the'interpolation coefiicient. Instead of the usual Taylors series (m L a i the series can be written as m.) re-m re)f'(x x+%w- This form of the series follows directly from (1) by writing X11, for x and 6x for 6x. From series (2) there is obtained which will be called the modified Taylors series, expressing f(x) in terms of f(xn) and f (x). In this series the terms after the first on the right hand side each involve 5x, which is an independent increment, and a derivative of the final output f(x) which as aforesaid thus becomes the interpolation coeflicient, that is it forms the coefiicients of the terms 6x which are used to yield interpolation.

The sine series becomes and it is instructive to re-write this as +sin ch The forms (4) and (4a) are those in which instrumentation is most easily carried out, since 6x, 6x 6x to as many terms as necessary, can be obtained by tapping transformers linearly, quadratically, cubically and so on and feeding them from one or other of two output-channels as required. It will -be:seen that it is comparatively simple to perform higher order interpolation, since'the interpolation coefiicients are available being merely sin x and cos x, and advantage may be taken of this to use larger intervals than would "otherwise be possible between the storediparticular values of f(x.).

"A "diagram of a simple sine cosinecoinputing apparatus using cubic interpolation is shownin Figure 2. Voltages are set up at the studs bobr b1- representative of successive particular values of say sin x while voltages representative of the corresponding particular values of cosine x appear at the studs c ,c 0,, which are connected to b0, b1, b2 as described above. The displacement of the low speed brushes 11 14 represents the instantaneous value of xn and that of the high speed brushes 15 18 represents the instantaneous value of 6x. The high speed brushes 15, 16 co-operate with the studs connected to taps on a linearly tapped potentiometer 19 to which is fed back inductively from the wind ing 20, a voltage representative of cos x. The last said voltage is fed back from the cosine output terminal of the apparatus, indicated at 21, to form an interpolation coefficient signal. The high speed brushes 17 and 18 similarly co-operate with studs connected to taps on a potentiometer 22 to which a voltage representative of sin x is fed back from the sine output terminal 23 of the apparatus to form another interpolation coefiicient signal. The output at the terminal 21 is obtained from the centre tap of the potentiometer 22 whilst the output at the terminal 23 is obtained from the centre tap of the potentiometer 19. The studs of the high speed brushes 15, 16 and 17, 18 correspond to successive values of 6x, the displacement of the brushes being representative of the instantaneous value of 5x, the taps on the potentiometers 19 and 22 being such that the voltage on each tap, with reference to the centre tap, is representative of 6x cos x or 6x sin x as the case may be where 6x has the value corresponding to the respective contact stud. Each lead to the taps on the potentiometer 19 and 22 receives further voltage from transformer devices 24 27 indicated in block form. The devices 24 and 27 are respectively arranged to inject into the leads voltages representative of sin x (derived by feedback from 23) multiplied by the values of 6x and 6x appropriate to the particular leads. Similarly the devices 25 and 26 inject respectively voltages representative of 6x cos x and 6x cos x. Thus, for example, the device 24 comprises a primary winding across which the voltage fed back to the terminal 23 is applied, and a series of secondary windings connected in series in the leads to the taps on the potentiometer 19, the number of turns in each secondary winding being representative of the value of 6x where 61: has the value corresponding to the stud at which the lead terminates. The devices 25, 26 and 27 may be similarly constructed. In this way the output fed to 23 is built up according to Equation 4 and similarly the output fed to 21 is built up according to the Equation 4a, sin x and cos x again, being the interpolation coefiicient signals.

An interesting and very practical application of the present invention is that of de-resolution of Cartesian coordinates into polar co-ordinates, in which, because of the Way in which sine and cosine are handled in combination, considerable simplification can be achieved.

Use is made of the basic equations =x cos S+y sin S (5) where R and the angle S are respectively the polar coordinates and x and y are the Cartesian co-ordinates. The voltage representing the right-hand side of (6) is used to drive a servomotor, the angular displacement of Whose shaft is used to set up the angle S. Two electrical inputs, x and y, are thus transformed to one mechanical output, S, and one electrical output, R.

In order to illustrate the potentialities of the system, the case of cubic interpolation will be considered.

Let S=0n+60Where 0n is the value of S at a reference point,.and the equivalent angle of the high'speed shaft. Also let 0=x sin S-y cos S U =x sin (i -y cos U=x sin (6,,+60) --y cos (0 +60) 3 U= U,,+R50+ g -2 but at balance U is zero so that where 60 and higher powers have been neglected.

It may be noted in passing that these last equations are equivalent to R =R cos 66 =-R sin 80 and they can easily be derived geometrically in this form.

A schematic diagram of an apparatus operating on the above principle is shown in Figure 3, and it will be observed that a considerable saving in complexity has in fact been achieved compared with the two units similar to Figure 2 which would otherwise be needed. The devices indicated in block form at 28 and 29 are arranged to set up x sin 0 x cos 0,, y sin 6 and y cos 0 as indidated, the devices each being similar to the slow speed part of Figure 2, except that instead of being energised with a fixed reference voltage, they are energised by voltages representative of x and y respectively. An output representative of R is taken from an output terminal denoted by the reference R and feedback from the output is applied to a transformer winding 30 via transformer windings 33 and 34 so that the feedback voltage applied to the Winding 30 is representative of the appropriate submultiple R, which in this case is R/2, the lower end (in the drawing) of winding 30 being connected as shown to the output terminal. The winding 30 is tapped quadratically and the taps are connected to contact studs of a high speed switch which has brushes 31 and 32 driven by the high speed shaft of the apparatus, the angular displacement of the brushes representing 60 and the taps on the winding 30 being such that the voltage set up between each tap and the lower end of the winding is representative of R60 /2, where 66 has the value corres'pohding to the respective stud. The brushes 31 and 32 arefasshown fed with a voltage respresentative of x cos 0 +y sin (i that is R from the units 28 and 29, and this voltage is fed via one or other of the studs of the high speed switch to the corresponding tap on the winding 30 and thence to the output terminal R, so that the output voltage R is equal to R +R60 /2 as required. The winding 35, forming part of the interpolating potentiomet'erof the U section of the apparatus, is fed with a voltage representative of R since it is inductively coupled to the primary winding 33 and it is linearly tapped, the taps being connected as indicated to the contact studs of a high speed switch which has brushes 36 and 37 driven by the high speed shaft of the apparatus. The angular displacement of the brushes 36 and 37 is thus representative of 60, an output representative of U is taken from the mid-point of the winding 35 and the voltage set up at each tap on this potentiometer is represent ative of R60, where '60 has the value corresponding to the respective stud to which the tap is connected. The transformer unit enclosed in the dotted rectangle 38 injects R50 /6 into each lead from the potentiometer 35, the value of 60 for each lead being appropriate to the displacement of the corresponding stud in the path of the brushes 36 and 37. Thus the transformer unit 38 hasa primary winding energized from the winding 50 on the R section transformer with a signal representative of R/ 6 whilst the secondary winding is connected in series in each lead, each secondary winding having a polarity and number of turns representative of 60 where 50 has the value corresponding to the respective stud. The brushes 36 and 37 are 'fed with voltages representative of x sin 0 -y cos 0 that is U from the units 28 and 29 and this is fed via one of the studs, and thence via a secondary winding of the transformer unit 38 and via a section of the winding 35 to the output point so that the output voltage U is equal to U1z+R30-R50 /6 as required. A servomotor 51 responsive to the output U drives the shafts of the high speed brushes in the units 23 and 29 until U is Zero. S is then represented by the position of the high speed shaft taken in conjunction with the position n of the low speed shaft in the units 28 and 29.

If the steps of the brushes 36 and 37 are too large to enable U to be reduced identically to zero in all cases, the first order term U60 in the expansion of R is not zero, but an appropriate voltage can be injected into the leads from the potentiometer 30 to the R switch 31, 32 by means of a further linear transformer unit fed voltage representing U. 7

Figure 4 illustrates a modification of Figure 3, in which such injection is made, corresponding parts in Figures 3 and 4 being denoted by the same reference numerals, and some parts being omitted in Figure 4 in the interests of simplicity. Two injection transformer units 41 and 42 are used each having secondaries in series with alternate leads from the tapped auto-transformer 30 and having primaries fed with signals representing U as will hereinafter appear. The U section high speed switch with brushes 36 and 37 of Figure 3 is replaced by two switches with brushes 45 and 46 and withtheir studs arranged to be electrically alternate, that is the studs under brush 45 correspond to alternate values of 60 while those under brush 46 correspond to the intervening values as indicated. These studs are fed from two secondary windings 39 and 40 which replace the winding 35 of the R transformer in Figure 3. The primary windings of the transformer units 41 and 42 receive their inputs (representative of U) from the centre taps of the windings 39 and 40 respectively. The arrangement of the switches is such that when a brush on the R switch (say the brush 31) is about to leave stud 43 for example, the voltage being injected into the lead from the winding 30 comes from the transformer 40 and stud 43 with which brush 46 is in contact. But already brush 45 will have made contact with stud 47 (assuming the switches rotate in the clockwise direction) and stud 44 will have an injected voltage fed to it appropriate to its own value of 60. This ensures that studs 43 and 44 and similarly other successive studs of the R switch have substantially identical voltages when the brush changes over from one to the next and will secure a smooth gradation of output voltages. A high speed switch 49 must now be employed to select the final U outputfrom the two values obtainable at the centre taps of transformer secondaries 39 and 40. As shown this additional switch 49 is providing additional interpolation but it could be employed merely as a change over switch.

The saving in complexity in the arrangements illustrated in Figures 3 and 4 has been achieved by the combination of sine and cosine at the reference point stage, and in the employment of R and U in Taylors series instead of their constituent parts. A still further simplification results from the very small or zero value of some of the differential coeflicients.

What we claim is:

1. Apparatus for evaluating the instantaneous value of an exponential function e where a is a constant and x is an independent variable, comprising means for storing values of the function for successive particular values of the independent variable x, switch means operable in dependence upon x for selecting the stored value of the function for the particular value of x nearest the instantaneous value thereof, further switch means operable in dependence on x to represent 6x the difference between the instantaneous value of x and said nearest particular value, means for forming an incremental signal proportional to the product of 6x and the selected value of the function, said means being predetermined to cause the constant of proportionality of the incremental signal to represent a and means for adding the selected value of the function and said incremental signal so as to evaluate approximately the function for the instantaneous value of x.

2. Apparatus for evaluating the instantaneous value of cos ax where a is a constant and x is an independent variable, comprising means for storing values of'the functions cos ax and sin ax for successive particular values of the independent variable x, switch means operable in dependence on x for simultaneously selecting the stored value of cos ax and sin ax for the particular value of x nearest the instantaneous value thereof, further switch means operable in dependence on x for forming the product of 06x and the selected value, to represent the difference 6x between the instantaneous value of x and said nearest particular value thereof, means for forming an incremental signal proportional to the product of 6x and the selected value of sin ax, said last means being predetermined to cause the constant of proportionality of the incremental signal to represent a and means for subtracting said incremental signal from the selected value of cos ax to evaluate approximately cos ax for the instantaneous value of x.

3. Apparatus for evaluating the instantaneous value of a function f(x), comprising means for storing values of f(x) for successive particular values of the independent variable x, switch means operable in dependence on x for selecting the stored value of (x) for the particular value of x nearest the instantaneous value thereof, means for setting up at least the first derivative of an output from said apparatus, means operable in dependence upon x to represent 5x the difference between said instantaneous value of x and said nearest particular value, means for forming an incremental signal proportional to the product of 6x and said first derivative, means for adding said incremental signal to the selected value of f(x) to ftgrm said output, whereby said output is representative 0 which is approximately the value of f(x) for the instantaneous value of x.

4. Apparatus for converting Cartesian co-ordinates x and y into polar coordinates R and S, comprising means for setting up the values of x cos 6, y sin 0, x sin 6 and 1 cos 6 for successive particular values of 0, switch means operable in dependence on 0 to select the values of x cos 0, y sin 0, x sin 0 and y cos 0 for the particular values of 0 nearest the instantaneous value thereof, means for adding the first and second-mentioned selected values to form a quantity R71, and for subtracting the fourth from the thirdmentioned selected values to form a quantity U11, means operable in dependence on 0 to represent 60 /2 where 60 is the difference between said particular value of 0 and the instantaneous value thereof, means for forming an incremental signal representing the product of /2 and a first output from the apparatus, means for adding said incremental signal to the quantity R11, to form said first output, further means operable in dependence on 0 to represent and 60 6, means for forming the product of 50 and said first output, means for forming the product of 69 6 and said first output, means for subtracting said last-mentioned product from said second last-mentioned product to form a second incremental signal, and means for adding said second incremental signal to U11. to form a second output from said apparatus, whereby said second output is approximately representative of the instantaneous value of x sin 0y cos 0 whilst said first output is approximately representative of the instantaneous value of x cos B-I-y sin 0, and has the value of R whilst 0 has the value S when 0 has such a value that said second-mentioned output is zero.

5. Apparatus according to claim 4, comprising means for forming the product of the second-mentioned output and 60, and means for subtracting the last-mentioned product from the sum of Rn and the first-mentioned product to form said first output thereby to increase the accuracy of the evaluation of R.

6. Apparatus for computing instantaneous values of a function of a variable, comprising value storage means for setting up signals representative of values of the function for particular values of the variable, selector means for selecting the signal representing the function for the particular value of the variable nearest the instantaneous value thereof, means for setting up a difference signal representing the difference between said nearest and instantaneous values of the variable, means for deriving an interpolation coefficient signal responsive to said selected signal, means for forming the product of said difference signal and said interpolation 'coefiicient signal, and means for adding said product to said selected signal to form an output representative approximately of the instantaneous value of said variable.

7. Apparatus for computing instantaneous values of a function of a variable, comprising value storage means for setting up signals representing values of said function for successive particular values of said variable, selector meansfor selecting from said storage means a signal representing the function for the nearest particular value of said variable, means for setting up a difierence signal representing the difference between said nearest and instantaneous values of the variable, an output circuit, means for deriving a signal representing a derivative with respect to said variable of the output from said output circuit, means for forming an incremental signal comprising the product of said derivative and a power of said difference signal, means for adding said incremental signal to said selected signal and for feeding the sum so formed to the output circuit, whereby the output from said output circuit approximately represents the instantaneous value of said function.

8. Apparatus for evaluating the instantaneous value of sin ax where a is a constant and x is an independent variable, comprising means for storing values of the functions cos ax and sin ax for successive particular values of the independent variable, switch means operable in dependence on x for simultaneously selecting the stored value of cos ax and sin ax for the particular value of x nearest the instantaneous value thereof, further switch means operable in dependence on x to represent the difference 6x between the instantaneous value of x and said nearest particular value thereof, means for forming an incremental signal proportional to the product 6x and a selected value of cos ax, said last means being prededetermined to cause the constant of proportionality of 9 the incremental signal to represent a and means for addsaid incremental signal to the selected value of sin ax to evaluate approximately sin ax fior the instantaneous value of ax.

9. Apparatus for conventing Cartesian coordinates x and y into polar coordinates R and S, comprising value storage means responsive to input signals representing instantaneous values of x and y for setting up signals representing respectively the functions x cos 0, y sin 0, x sin and y cos 0 for successive particular values of 0, selector means for selecting the value of each of said functions for a particular value of 0, means for utilizing said selected values to form the quantity x cos 6+y sin 0 and the quantity x sin 0y cos 0, means for forming a difference signal representing the difierence between said particular value of 0 and the value of 0 which makes said second sum zero, first and second output circuits, means for forming incremental signals each comprising a prodnot of the output from one of said output circuits and a power of said difierence signal, means for adding incremental signais respectively to said first and second quantities and for feeding the total including said first quantity to said first output circuit and for feeding the total including said second quantity to said second output circuit to make the output of said second output circuit zero and that of said first circuit approximately representative of R, whereby 0 with the addition of said difierence signal approximately equals S.

10. Apparatus fior evaluating cos x and sin x where x is a variable angle, comprising means for storing values of cos x and sin x ior successive particular values of x, switch means operable in dependence upon x for selecting from said first means values of cos x and sin x for the particular value of x nearest the instantaneous value thereof, further switch means operable in dependence on x to represent 6x, the difierence between the instantaneous value of x and said nearest particular value, means for forming a first incremental signal proportional to the product of a first output from said apparatus and x, means for forming a second incremental signal proportional to the product of a second output from the apparatus and 6x, means for subtracting said first incre- 10 mental signal from the selected value of cos x to form said second output whereby said second output approximately represents the value of cos x for the instantaneous value of x, and means for adding said second incremental signal to the selected value of sin x to form said first output whereby said first output approximately represents tlie value of sin x for the instantaneous value of x.

11. Apparatus for computing values of two functions which are so related that each function, apart from con stant terms, is the positive of alternate derivatives of the other function, comprising value storage means for setting up signals representing values of the two related functions for particular values of the variable of the functions, selector means for selecting from said storage means a first signal representing one function for the particular valueof the variable nearest the instantaneous value thereof, selector means for selecting from said storage means a second signal representing the other finnction for the particular value of the variable nearest the instantaneous value thereof, means responsive to a first output of said apparatus for deriving a first incremental signal, means responsive to a second output of said apparatus for deriving a second incremental signal, means for adding said first signal and said second incremental signal to evaluate approximately the first function for the instantaneous value of the variable and thereby form said first output, and means for adding said second signal and said first incremental signal to evaluate approximately the second function for the instantaneous value of the variable and thereby form said second output. 7

References Cited'in the file of this patent UNITED STATES PATENTS 2,404,387 Lowell et al. July 23, 1946 2,507,890 Orowtner Nov. 19, 1947 2,436,178 Rajchman Feb. 17, 1948 2,634,909 Lehmami Apr. 14, 1953 FOREIGN PATENTS 600,896 Great Britain Apr. 21, 1948 

